Angles Formed by Parallel Lines and Transversals

You are studying in any school grade !!! Then remember you can never get rid of yourself, Angles/Parallel Lines/Transversal concepts are used in most of the chapters of mathematics. Even in various competition exams after completing school you will get questions on this topic. Let's discuss transversal first.

Concept of Transversal - Transversal Meaning

A straight line which cuts two or more given straight lines is called a transversal.

In the adjoining figure, PQ cuts straight lines AB and CD, and so it is a transversal.

When a transversal cuts two given straight lines, the following pairs of angles are formed.

Transversal Lines and Angles- Examples

1. Two pairs of interior alternate angle:

Angle marked 1 and 2 form one pair of interior alternate angles, while angle marked 3 and 4 form another pair of interior alternate angle.

2. Two pairs of exterior alternate angle:

Angles marked 5 and 8 form one pair, while angle marked 6 and 7 form the other pair of exterior alternate angles.

3. Four pairs of the corresponding angle:

Angle marked 3 and 6, 1 and 5, 8 and 2, 7 and 4 form four pairs of corresponding angles.

4. Two pairs of allied or co-interior or conjoined angle:

Angle marked 3 and 2 form one pair of allied angle and angle marked 1 and 4 form another pair of allied angles.

Parallel Lines and Angles

Two straight lines are said to be parallel, if they do not meet anywhere, no matter how long they are produced in any direction.

The adjacent figure shows two parallel lines AB and CD.

When two parallel lines AB and CD are cut by a transversal PQ:-

alternate interior angles definition geometry

Angles which makes Z-SHAPE are called alternate angles.

1. Interior and exterior alternate angles are equal:

i.e.∠ 3 = ∠6 (interior alternate)

∠ 4 = ∠ 5(interior alternate)

∠1 = ∠8((exterior alternate )

∠2 = ∠ 7(exterior alternate )

Corresponding Angles

2. Corresponding angles are equal:

Angles lying on the same side of the transversal are called corresponding angles.

i.e. ∠ 1 = ∠5, ∠ 2 = ∠6, ∠3 = ∠7 and ∠ 4 = ∠ 8.

3. Co-interior or allied angles are supplementary:

i.e. ∠ 3 + ∠ 5=180 and ∠4 + ∠6 =180.

Co-interior angles make H-SHAPE. Or we can say a closed box open at one end.

Conditions of Parallelism

If two straight lines are cut by a transversal such that:

1. A pair of alternate angles are equal, or

2. A pair of corresponding angles are equal, or

3. The sum of interior angles on the same side of transversal is 180, then two straight lines are parallel to each other.

Therefore, in order to prove that the given lines are parallel; show either alternate angles are equal or, corresponding angles are equal or, the co-interior angles are supplementary.